How Do You Calculate the Effective Speed of a Tailwind for a Westward Cyclist?

[Coco12]

Homework Statement

A cyclist head due west on a straight road at 5.6m/s. A northeast wind is blowing at 10m/s. What is the effective speed of the tailwind?(resultant)

Homework Equations

Cos 45 10
Sin 45 10

The Attempt at a Solution

Basically I broke it down into its x and y components. Then added them together to the Rx and Ry and used Pythagorean to find the resultant. I just have one question though: when adding the x components for the wind and the cyclist speed. Would the Rx be -5.6m/s + 7.1( 7.1 is derived from cos 45 degrees *10) = 1.5? Or would the 5.6 be positive?

 

[CAF123]

Would the Rx be -5.6m/s + 7.1( 7.1 is derived from cos 45 degrees *10) = 1.5? Or would the 5.6 be positive?

To help you answer your question, it might help to write out explicit expressions for the vectors representing the cyclist and the wind velocities.

 

[Coco12]

To help you answer your question, it might help to write out explicit expressions for the vectors representing the cyclist and the wind velocities.

I did. Rx=ax+bx
Ry=ay +by

I'm just wondering since it's 5.6 m/s due west which on a Cartesian plane would be a negative x whether I would incorporate the negative when trying to find rx

 

[CAF123]

I did. Rx=ax+bx
Ry=ay +by

I'm just wondering since it's 5.6 m/s due west which on a Cartesian plane would be a negative x whether I would incorporate the negative when trying to find rx

Yes, you can represent the velocity vector of the cyclist as $\vec{C} = -5.6 \hat{x}$ and that of the wind as $\vec{W} = (10 \cos 45)\hat{x} + (10 \sin 45) \hat{y}$. Now, as you said, just add components to get the resultant.

From a more conceptual point of view, imagine a game of tug of war. Person A pulls to left with force 5.6N and person B to right with force 10cos45 ≈ 7.1 N. The person pulling to right wins, but only marginally. (winning by 7.1 - 5.6, not 7.1 + 5.6)

 

[Nickie]

The Rx would be positive, as the cyclist is traveling in the same direction as the wind, so their velocities would add together. Therefore, the effective speed of the tailwind would be 5.6m/s + 7.1m/s = 12.7m/s. It is important to pay attention to the direction of the vectors when adding them together.